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The chromatic polynomial of grid graph \(P_3 \square P_n\)

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Abstract

This paper studies the chromatic polynomial of some special graphs like path graph, ladder graph and grid graph using the concept of digonalization of transfer matrix to solve simultaneous recurrence relation. The aim of this paper is to solve an open problem on the chromatic polynomial of grid graph (Problem 8.1 of Selected Topics in Graph Theory, Volume 3 which is edited by Lowell W. Beineke and Robin J. Wilson) and give general formula of the chromatic polynomial in \(\lambda \) of grid graph \(P_3 \square P_{n}\), where \(n \in \mathbb {N}\) without any condition on n where \(\lambda \) denotes number of available colors.

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References

  1. Birkhoff, G.D.: A determinant formula for the number of ways of coloring a map. Ann. Math. 14, 42–46 (1912)

    Article  MathSciNet  Google Scholar 

  2. Whitney, H.: The colorings of graphs. Ann. Math. 33, 688–718 (1932)

    Article  MathSciNet  Google Scholar 

  3. Read, R.C.: An introduction to chromatic polynomials. J. Comb. Theory 4(1), 52–71 (1968)

    Article  MathSciNet  Google Scholar 

  4. Biggs, N.: A matrix method for chromatic polynomials. J. Comb. Theory Ser. B 82(1), 19–29 (2001)

    Article  MathSciNet  Google Scholar 

  5. Dong, F.M., Koh, K.M., Teo, K.L.: Chromatic Polynomials and Chromaticity of Graphs. World Scientific, Singapore (2005)

    Book  Google Scholar 

  6. Thomassen, C.: The chromatic polynomial and list colorings. J. Comb. Theory Ser. B 99(2), 474–479 (2009)

    Article  MathSciNet  Google Scholar 

  7. Read, R.C., Tutte, W.T.: Chromatic polynomials. Sel. Top. Graph Theory 3, 15–42 (1988)

    MathSciNet  Google Scholar 

  8. Srinivas, B.R., Chaitanya, A.S.K.: The chromatic polynomials and its algebraic properties. Int. J. Sci. Innov. Math. Res. 2(11), 914–922 (2014)

    Google Scholar 

  9. Nizami, A.R., Munir, M., Khan, A.S., Shahzadi, Z.: On chromaticity of ladder-type graphs. Sci. Int. 28(2), 00–00 (2016)

    Google Scholar 

  10. Baxter, R.J.: Chromatic polynomials of large triangular lattices. J. Phys. A Math. Gen. 20(15), 5241 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  11. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs. CRC Press, Boca Raton (2011)

    Book  Google Scholar 

  12. Weisstein, E.W.: Chromatic Polynomial. From MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/ChromaticPolynomial.html

  13. Lowell, W.B., Robin, J.W.: Selected Topics in Graph Theory, vol. 3. Academic Press, London (1978)

    Google Scholar 

  14. Perrett, T.: Chromatic Polynomial Calculator. GitHub, United States (2021). https://tjperr.github.io/chromatic-polynomial/

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Acknowledgements

This work was supported by CSIR, New Delhi under senior research fellowship CSIR-Award No: 09/382(0229)/2019-EMR-I through the first author.

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Author notes

  1. Renu Yadav, Amit Sehgal, Sarita, and Archana Malik have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana, 124001, India

    Renu Yadav & Archana Malik

  2. Department of Mathematics, Pandit Neki Ram Sharma Government College, Rohtak, Haryana, 124001, India

    Amit Sehgal

  3. Department of Mathematics, Government (Post Graduate) College for Women, Rohtak, Haryana, 124107, India

    Sarita Sehgal

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  1. Renu Yadav
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  2. Amit Sehgal
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Correspondence to Renu Yadav.

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Yadav, R., Sehgal, A., Sehgal, S. et al. The chromatic polynomial of grid graph \(P_3 \square P_n\). J. Appl. Math. Comput. 70, 619–637 (2024). https://doi.org/10.1007/s12190-023-01967-4

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  • DOI: https://doi.org/10.1007/s12190-023-01967-4

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